Which Factors for Corporate Bond Returns? (2024)

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Abstract

A pivotal issue in finance is understanding why certain types of assets, on average, earn vastly different returns than others do. Researchers and practitioners often attempt to explain these returns with factor models that consist of a sparse set of factors. In equity markets, hundreds of factors have been proposed, and equity managers have applied factor investing successfully for decades.1 Factor investing in corporate bonds, on the other hand, is a relatively unexplored field. However, searching for bond factors has recently attracted growing interest, and, based on these discoveries, factor investing is likely to pick up substantially in the coming years.

A plethora of factors lead to the necessity from both an academic and a practitioner’s perspective to know which are genuine risk factors in corporate bond markets that provide incremental information about returns. In this paper, we thus address the following questions: Do we really need all factors proposed in the corporate bond literature to explain the cross-section of returns? Which factors move corporate bond prices systematically? What set of factors overall best describes corporate bond returns? Are some factors redundant relative to others? To what extent does each needed factor play a role in explaining time-series and cross-sectional variation in corporate bond returns? Which economic forces drive the factors?

Our main contribution is a systematic analysis of the factors proposed in the corporate bond pricing literature. Our study helps academics and practitioners separate useful factors from redundant ones and search the growing list of bond factors for a set that spans the tangency portfolio and collectively best explains the differences in corporate bond returns. Based on this, we can build an “optimal” corporate bond factor model. To the best of our knowledge, we are the first to comprehensively compare a broad set of common and recently proposed factors and factor models for corporate bonds.

We start our empirical analysis by considering a collection of, from our point of view, the 23 most prominent risk factors in the corporate bond literature. We use the bond market (MKTb), term (TERM), and default risk (DEF) (Fama and French 1993), credit risk (CRF), downside risk (DRF), liquidity risk (LRF), and short-term reversal (STR) (Bai, Bali, and Wen 2019), momentum (MOMb) and long-term reversal (LTR) (Jostova et al. 2013; Bali, Subrahmanyam, and Wen 2017, 2021a), bond volatility (BVL), carry (CRY), duration (DUR), stock momentum (MOMs), and value (VAL) (Israel, Palhares, and Richardson 2018; Kelly, Palhares, and Pruitt Forthcoming), economic uncertainty (UNC) and (tax) policy uncertainty (EPU, EPUtax) (Bali, Brown, and Tang 2017; Bali, Subrahmanyam, and Wen 2021b; Tao et al. 2022; Lee 2022), and volatility risk (VOL) (Chung, Wang, and Wu 2019), along with the five Fama and French (2015) stock market factors (MKT, SMB, HML, RMW, CMA) (Bektić et al. 2019).

Given the apparent importance and need for replicability of factor premiums, as highlighted by a growing number of meta-studies, for instance, Welch and Goyal (2008), Harvey, Liu, and Zhu (2016), McLean and Pontiff (2016), Green, Hand, and Zhang (2017), Linnainmaa and Roberts (2018), and Hou, Xue, and Zhang (2020), we examine these published factors on the same pedestal using the same period, data sources, and bond return definitions. While the choice of alternative specifications and procedures is not technically wrong, using factors that are constructed consistently helps us to avoid comparing apples with oranges.

In the first part of our empirical study, we use the necessary condition of the factor identification protocol popularized by Pukthuanthong, Roll, and Subrahmanyam (2019). With this step, we basically separate factor candidates that systematically move corporate bond prices from those that do not. A factor candidate cannot be a viable risk factor if it does not move prices. Technically, we analyze whether the factor candidates can explain the canonical correlations between the entire set of factors and test asset principal components. We only retain those factors that pass the identification protocol for further analysis. We find that many prominent factors already fail this first test. For example, all the factors proposed by Bai, Bali, and Wen (2019) do not satisfy the necessary condition for being a risk factor in corporate bond markets. In addition, the Tao et al. (2022) and Lee (2022) policy uncertainty factors, the Israel, Palhares, and Richardson (2018) value factor, the short-term reversal factor, and all Fama and French (2015) stock market factors are eliminated.

As a second step, we employ the Bayesian marginal likelihood model comparison method recently developed by Barillas and Shanken (2018) and Chib, Zeng, and Zhao (2020) (BS-CZZ). The key advantages of this approach are that (a) it enables us to simultaneously compute the model probabilities for the collection of all possible models that are subsets of the given factors, while (b) it takes into account the matter of parsimony. The first main result is that a four-factor union of carry, duration, stock momentum, and term structure factors is revealed by the data as the best (no. 1) corporate bond risk factor model in terms of the Bayesian posterior probability. Based on a Bayes factor cutoff, only three further models remain as serious contenders. All four winning models contain the carry and stock momentum factors. The duration and term-structure factors have cumulative posterior probabilities around 50%.

To provide direct statistical evidence on the relative performance of different models, we use the Barillas et al. (2020) test of the equality of squared Sharpe ratios. We conduct pairwise comparisons among the winning models and various existing models. We find that the no. 1 winning factor model yields a substantially and significantly larger squared Sharpe ratio than all other contenders. Thus, the selected set of factors clearly dominates the existing models. We show that this is not only true in-sample. Also, out-of-sample, with two different sample splitting schemes, the winning models generate the highest Sharpe ratios.

We continue our analysis by running two sets of spanning tests. The purpose of the first set is to demonstrate why the various existing models fall short of explaining the winning factors. We find that the existing models largely fail to explain the average returns of the carry and stock momentum factors. However, even the Israel, Palhares, and Richardson (2018) and Kelly, Palhares, and Pruitt (Forthcoming) models that contain one or both of these factors are rejected by the Gibbons, Ross, and Shanken (1989) (GRS) test. On the other hand, the no. 1 winning model can explain all other factors that pass the first-step identification protocol.

In a penultimate step, we use time-series and cross-sectional asset pricing tests with test assets to thoroughly analyze the performance of all models. We find that the winning models perform reasonably well for these tasks. In the time-series tests, the four winning models, along with those of Israel, Palhares, and Richardson (2018) and Kelly, Palhares, and Pruitt (Forthcoming), which share many of the same factors, typically yield the smallest GRS test statistics, the lowest average absolute alphas, and the smallest squared Sharpe ratios of the alphas for different sets of test assets. In cross-sectional tests, the same set of models performs best. We find that the no. 1 winning model yields the largest cross-sectional R²s.

Finally, we examine the economic drivers of the corporate bond risk factors contained in the winning models. We find that corporate bond illiquidity along with volatility are important drivers of the carry and duration factors. The duration factor, however, is also strongly driven by intermediary distress and inflation. The main determinant of the stock momentum factor is inflation, while the term factor is mainly driven by the change in industrial production.

The findings in this paper have important practical implications. The winning set of factors can be used as a benchmark model for future research and in performance evaluation. Furthermore, investors in corporate bond markets can build on our findings to implement the most promising factor-investing strategies.

1. Literature Review

Both stocks and corporate bonds are contingent claims on the value of the same underlying firm. However, several notable features distinguish bond from stock markets, suggesting potential market segmentation. Indeed, Chordia et al. (2017) and Choi and Kim (2018) find a discrepancy in risk premiums between corporate bond and equity markets. Therefore, it is important to investigate the cross-section of corporate bond returns by also using the factors constructed based on corporate bond characteristics, rather than only relying on the available commonly used factors from the equity market. This direction also helps to facilitate factor-based investing strategies in corporate bond markets. Hence, we focus our study mainly on corporate bond factors.

Earlier studies, notably Fama and French (1993), generally utilize aggregate bond indexes, such as the term and default spread factors, to explain the cross-sectional variation in corporate bond returns. Recently, inspired by the way characteristics have been used for constructing equity factors, substantial research efforts have been devoted to exploring new factors that drive corporate bond returns. Bali, Subrahmanyam, and Wen (2017) and Bali, Subrahmanyam, and Wen (2021a) examine whether short-term reversal, momentum, and long-term reversal are priced in the corporate bond market. Bali, Subrahmanyam, and Wen (2017) introduce a return-based factor model including three factors constructed based on the bond market factor and these past return characteristics. Bai, Bali, and Wen (2019) propose a four-factor model, including the bond market as well as factors that build on the downside risk, credit risk, and liquidity risk characteristics, which appear to be prevalent in the corporate bond market. Israel, Palhares, and Richardson (2018) and Kelly, Palhares, and Pruitt (Forthcoming) propose alternative factors and factor models based on the bond volatility, carry, duration, stock momentum, and value characteristics. Bektić et al. (2019) study the Fama and French (2015) five-factor model in corporate bond markets. Chung, Wang, and Wu (2019), Bali, Subrahmanyam, and Wen (2021b), Tao et al. (2022), and Lee (2022) find that aggregate volatility and economic and policy uncertainty are priced in the cross-section of corporate bond returns. In this study, we comprehensively examine the properties of the factors introduced in these studies (in total 23) and form an optimal factor model.

Two competing approaches demonstrate the ability of factor models in explaining the cross-section of returns: left-hand-side (LHS) and right-hand-side (RHS) approaches, as classified by Fama and French (2018). LHS approaches introduce additional test assets and examine the models based on their abilities to price these test assets. For these, it often comes down to alphas, which are the estimated intercepts from time-series regressions of base asset returns on these factor models. Alphas capture the difference between the return an asset actually earns and what a factor model would predict, and hence gauges the model’s error. A model with lower average pricing errors is deemed to perform better. Empirical implementations using characteristic/industry-sorted portfolios as the LHS test assets and assessing model performance by alpha-based statistics from time-series regressions to explain the LHS assets are ubiquitous. In numerous studies in equity markets, such as Fama and French (2015, 2018), Hou, Xue, and Zhang (2015), or Stambaugh and Yuan (2017), competing factor models are evaluated using several comparative alpha-based criteria, for example, the number of significant alphas based on t-statistics, the number of rejections by the GRS test, the point estimates of average absolute alpha, and the average absolute t-statistic. Among others, Bali, Subrahmanyam, and Wen (2017) and Bai, Bali, and Wen (2019) also aim to identify a superior model for corporate bond returns as the one that generates a smaller average absolute alpha and delivers a larger average time-series regression R² for certain test assets.

While the LHS alpha-based setting appears frequently in the empirical literature, some criticize it as being problematic. First, the selection of test assets is not innocuous. One important critique about standard asset pricing tests, brought forward by Lewellen, Nagel, and Shanken (2010), is that characteristic-sorted portfolios used as test assets do not have sufficient independent variation in the loadings of factors constructed with the same characteristics. Second, this framework ignores the pricing impact of factors from other models. Third, Barillas and Shanken (2017), Fama and French (2018), and Ahmed, Bu, and Tsvetanov (2019) show in numerous examples that informally comparing point estimates of alpha-based performance metrics may yield inconsistent model rankings, which can lead to incorrect judgments on the pricing performance of different models.

Barillas and Shanken (2017) straightforwardly emphasize that models should be judged in terms of their power to explain not only test assets but also the traded factors in other models (ideally, the entire universe of returns). They argue that LHS test portfolios do not provide any further information about model comparisons beyond what can be obtained by examining how well each model prices the factors of other models. Thus, following their argument, test assets are irrelevant for the purpose of model comparison.

This revealing insight leads to the development of the so-called “RHS approach.” Here, spanning regressions only involve other factors regressed on those of a model in order to decide whether candidate factors add explanatory power to a benchmark model. If the intercept is zero, the candidate factors contribute no additional information. Spanning tests are the main method adopted by Hou et al. (2019) and Daniel, Hirshleifer, and Sun (2020) to compare factor models.

Barillas and Shanken (2018) develop a Bayesian RHS setting that permits us to compare a large set of models simultaneously and identify the best, parsimonious one. There are also alternative recent (LHS) approaches for model selection (e.g., Feng, Giglio, and Xiu 2020; Hwang and Rubesam 2020; Harvey and Liu 2021). We opt for the BS-CZZ approach because it is economically motivated for exactly the task we intend it for: to find an optimal factor model.2 In addition, the approach turns out to be reasonably powerful, and the results of this RHS approach hold up well under an alternative LHS evaluation.3

Taking all the issues discussed above into consideration without being dogmatic on the LHS/RHS debate, the first part of this paper is based on an RHS approach, in which we scan for the best corporate bond factor model using a Bayesian marginal-likelihood-based method. Afterward, we analyze the winning models further by comparing their performance to existing models using two RHS approaches. Finally, we use multiple sets of LHS test portfolios to analyze the performance of the winning models and the existing ones for explaining cross-sectional and time-series variation in corporate bond returns.

2. Data and Methodology

2.1 Corporate bond data

We use the corporate bond data set of Kelly and Pruitt (2022). It is compiled from four sources: the Trade Reporting and Compliance Engine (TRACE) Enhanced, the Mergent Fixed Income Securities Database (FISD), Compustat, and the Center for Research in Security Prices (CRSP). Corporate bond transaction data (intraday clean price and volume) are from TRACE Enhanced and bond characteristics, such as bond ratings or coupons, are from FISD. Additional equity characteristics are from Compustat and CRSP. Special types of bonds, such as convertible bonds, bonds with floating coupon rates, and callable bonds are excluded from the data set.

The monthly return of corporate bond i in month t is calculated as

where $Pi,t$ is the clean transaction price, $AIi,t$ is the accrued interest, and $Ci,t$ is the coupon payment, if any, of bond i in month t.4

2.2 Candidate factors and models

In Table1, we briefly summarize the definitions of the bond variables (panel A) as well as the 23 candidate factors used in this study (panel B). More details on the exact construction of all variables and factors can be found in Appendix A.5

In addition to the individual factors, for comparison, we also consider a set of existing factor models. We indicate the set of factors included in the models in braces. That is, we consider a corporate bond CAPM (CAPMbond: {MKTb}), the Fama and French (1993) three-factor model for corporate bonds (FF3: {MKTb, TERM, DEF}), the Fama and French (1993) three-factor model for corporate bonds augmented by a liquidity risk factor and a bond momentum factor (aug. FF3: {MKTb, TERM, DEF, LRF, MOMb}), the Fama and French (1993) five-factor model for corporate bonds (FF5stkb: {MKTs, SMB, HML, TERM, DEF}), the Bai, Bali, and Wen (2019) four-factor model (BBW: {MKTb, DRF, CRF, LRF}), the four-factor model in the spirit of Bali, Subrahmanyam, and Wen (2017) (BSW: {MKTb, STR, MOMb, LTR}), the five-factor model in the spirit of Israel, Palhares, and Richardson (2018) (IPR: {CRY, DUR, MOMb, MOMs, VAL}), and the observable five-factor model of Kelly, Palhares, and Pruitt (Forthcoming) (KPP: {MKTb, CRY, DUR, BVL, VAL}).

2.3 First step: Factor identification protocol

For a first-step screening, we use the necessary condition of the factor identification protocol of Pukthuanthong, Roll, and Subrahmanyam (2019). The goal of this step is to identify factors that systematically move corporate bond prices. That is, the factors should be related to the covariance matrix of corporate bond returns.

As representative test assets for this step, we use a set of 12 industry portfolios, 25 size-maturity portfolios, 25 rating-maturity portfolios, and further $5×5$ double-sorted portfolios on the bond rating and 29 corporate bond characteristics provided by Kelly and Pruitt (2022). This set of portfolios clearly satisfies the requirements of Pukthuanthong, Roll, and Subrahmanyam (2019) that the test assets should belong to different industries and have sufficient heterogeneity. From these portfolios, we extract the first 10 principal components using the method of Connor and Korajczyk (1988).6 To account for possible nonstationarity, we cut our sample into two halves and do the analysis separately for each subperiod (Pukthuanthong, Roll, and Subrahmanyam 2019). Next, we calculate the canonical correlations between the candidate factors and these 10 principal components. Finally, we regress each of the 10 canonical variates (which are all weighted averages of the 10 principal components) on a constant and the set of factor candidates. As in Pukthuanthong, Roll, and Subrahmanyam (2019), for an eligible factor we require an average of the absolute t-statistics associated with the significant canonical correlations exceeding 1.96 and the average number of single absolute t-statistics exceeding 1.96 has to be higher than 2.5.

All factors that do not pass this first test apparently do not move bond prices and can be rejected as viable risk factors. Hence, we will only consider candidate factors that pass this factor identification protocol for the next steps.

2.4 Second step: BS-CZZ model comparison procedure

Among the factors that move corporate bond prices, we next aim to find those that best price the cross-section. We employ the Bayesian marginal-likelihood-based model comparison approach introduced by Barillas and Shanken (2018) and revisited by Chib, Zeng, and Zhao (2020).7 This method allows us to simultaneously compare all possible models based on the subsets of the given factor space. To scan for the best model, we compute their log marginal likelihoods to perform the prior-posterior update and then rank them based on their posterior probabilities.

In more detail, starting with a set of K (traded) potential risk factors, in general $J=2K−1$ factor combinations are possible. With the factor set resulting from the first-step factor identification and the restrictions on correlated factors described in Section 3.2, 1,024 candidate factor models remain. The model space is thus $M={M1,M2,…,MJ}$⁠. $Mj$ is one possible model defined by the vector of included factors $f˜j$ and that of excluded factors $fj*$⁠.

Each of the 1,024 factor models thus has a $Lj×1$ vector of included factors $f˜j$ and a $(K−Lj)×1$ vector of excluded factors $fj*$⁠. The data generating process of model j is thus given by

and

$α˜j$ is a $Lj×1$ parameter vector and $ϵ˜j,t$ is a multivariately normally distributed residual vector. $Bj,f*$ is a $(K−Lj)×Lj$ parameter matrix. $ϵj,t*$ is also a multivariately normally distributed residual vector. A special case applies when all factors are included in f$j$⁠.8

The log marginal likelihood of a model $Mj(j≠J)$ with y given the sample data of the factors over T time periods in closed form is

We provide the details on the computation of the terms in Equation (4) in Appendix B.

The end product of the scanning procedure is a ranking of models

by

where $M1*$ denotes the winning model, identified as the one that has the highest posterior model probability. Since the remaining terms in the posterior-probability calculation can be summarized by just a normalization constant, the ranking of posterior probabilities is equivalent to that of marginal livelihoods $m˜(y|Mj)$⁠.

2.5 Model comparison based on squared Sharpe ratios

After having determined the top model(s), the next step is a comparison to existing ones. For this purpose, among others, we use the Sharpe-ratio-based approach of Barillas et al. (2020) that requires a series of tests. First, we compute the differences between the bias-adjusted sample squared Sharpe ratios for various pairs of factor models.9 Second, we calculate the p-values for the test of equality of the squared Sharpe ratios in two cases of nested models and non-nested models.

In the case of nested models (i.e., all of the factors in one model are included in the other model), in order to determine whether the model with more factors is superior, we check whether the squared Sharpe ratio of the larger model is higher than that of the model with fewer factors. This is a test of whether alphas of the noncommon factors in the larger model (i.e., that are not contained in the smaller model) regressed on the smaller one are significantly different from zero, which can be done simply with the GRS test.

In the case of non-nested models (i.e., each model contains factors not included in the other model), the statistical analysis is a sequential test. The preliminary step entails comparing the squared Sharpe ratios of the model composed of all the factors from both models and the smaller one that contains only the common factors. It becomes equivalent to testing the null hypothesis that the alphas of the nonoverlapping factors on the common ones are zero. If this test fails to reject, then the evidence is consistent with the notion that the common factors model is as good as the models that add the noncommon factors. If this test is rejected, some or all of the noncommon factors are not redundant and contribute to an increase in the squared Sharpe ratio compared to the common factors model. However, we still do not know which non-nested model has a higher squared Sharpe ratio. Therefore, we then proceed with a direct test of the equality of the squared Sharpe ratios from the two non-nested models by calculating the p-value based on the results in Proposition 1 of Barillas et al. (2020).10

3. Model Selection

3.1 Summary statistics

Our sample includes 8,759 U.S. corporate bonds issued by 1,220 unique firms with 443,485 bond-month return observations in total during the sample period from July 2002 to December 2019. Over the sample period, on average, 83.85% of our rated bond sample are investment grade and 16.15% are noninvestment grade.11 On average, our sample includes approximately 5,361 bonds per month over the whole period.

Panel A of Table2 reports the descriptive statistics of our bond sample. The average monthly bond return is 0.50%, with a standard deviation of 2.17%. The sample contains bonds with an average rating of 8.02 (i.e., BBB+), and an average amount outstanding of $913 million. The average corporate bond in our sample is 3.44 years old, has a time-to-maturity of 8.57 years, and a duration of 5.92 years. The average corporate bond return, its distribution, as well as the other summary statistics are very similar to those reported in other studies (e.g., Jostova et al. 2013; Bai, Bali, and Wen 2019; Bali, Subrahmanyam, and Wen 2021a; Kelly, Palhares, and Pruitt Forthcoming).

Panel B of Table2 presents the summary statistics of the monthly factor returns between August 2003 and December 2019. Since a certain amount of data is first necessary to obtain the measures, the time series of DRF, CRF, EPU, EPUtax, LTR, STR, UNC, and VOL returns start somewhat later. LTR is the last factor with data available and starts in August 2006. For all tests, including LTR, we use the common sample period from August 2006. To place all factors on equal footing, we follow the approach of Bai, Bali, and Wen (2019), who are inspired by the classical Fama and French (1993) approach to equity factors, and obtain the factors via double-sorts with credit ratings. This way, we ensure that the factors genuinely pick up the risk and return related to their underlying economic variables rather than just passive exposure to credit risk.

The average bond market excess return for our sample is 0.34% per month and highly statistically significant with a t-statistic of 3.35. The average return is very similar to the 0.39% per month reported by Bai, Bali, and Wen (2019) for a slightly shorter sample period. The corporate bond factors mostly yield significantly positive average returns that are consistent with the previous literature.12

3.2 Factor identification results

We begin the empirical analysis with the factor identification protocol step of Pukthuanthong, Roll, and Subrahmanyam (2019). In panel A of Table3, we show the results for the canonical correlations between the 10 principal components extracted from the large set of test assets and the 23 candidate factors. We do this analysis for the two equal halves of our sample period. We find that in both halves 9 of the 10 canonical correlations are statistically significant. Thus, several pairs of canonical variates between the test assets and factors, with each being orthogonal to the others, have substantial intercorrelations.

The main output of the factor identification protocol step is in panel B of Table3. The bond market factor reaches an average t-statistic in the multiple regressions to explain the significant canonical variates of 4.9. In both subperiods, 7 of the 10 t-statistics for the bond market are statistically significant. Thus, MKTb clearly passes the hurdles set by Pukthuanthong, Roll, and Subrahmanyam (2019) for the factor-identification-protocol step. Other prominent corporate bond factors, however, fail this step. For example, the Bai, Bali, and Wen (2019) CRF, DRF, and LRF factors are eliminated. They are not sufficiently strongly related to the significant canonical variates of the test asset principal components. Hence, they do not appear to sufficiently strongly and systematically move corporate bond prices. Similarly, the factor identification protocol step eliminates STR from consideration.

In total, 12 of the 23 candidate factors are eliminated by this step. Quite intuitively, this step eliminates all five Fama and French (2015) equity factors. The factors being kept for further consideration include {MKTb, BVL, CRY, DEF, DUR, LTR, MOMb, MOMs, TERM, UNC, VOL}. Next, we aim to form an optimal factor model from a subset of these factors.

Before doing so, we should have a look at the correlations of the factors surviving the first-step factor identification. We present these correlations in panel C of Table3. Many factors are moderately correlated. Part of the factors have positive correlations in excess of 0.4 with the aggregate bond market: BVL, CRY, DEF, DUR, and TERM. On the other hand, LTR, MOMb, MOMs, UNC, and VOL have rather small correlations with the aggregate bond market. Among the other factors, we find that TERM and DEF are negatively correlated, consistent with Fama and French (1993). BVL, CRY, DEF, and DUR are also positively correlated among one another. MOMb and MOMs have negative correlations with most other factors. Finally, LTR, UNC, and VOL are not strongly correlated to most other factors.

The highest correlations are between the set of factors {MKTb, BVL, DUR}, each of which exceeds 0.8. It is thus likely that these factors capture similar economic risk sources (Gospodinov and Robotti 2021). Hence, going forward we will not consider models that include more than one of these factors.

The fact that the surviving factors apparently drive systematic movements in corporate bond returns and that most of them yield a statistically significant average return indicates that they could all be useful for pricing corporate bonds. The substantial correlations among different factors, on the other hand, suggest that the factors are not all that different. Thus, some factors are likely redundant and a parsimonious optimal factor model does not need all of them.

3.3 Model selection results

In a next step, we thus use the model selection approach to form an optimal model out of the 11 candidate factors. That is, we perform the second model selection step using the approach of Barillas and Shanken (2018) and Chib, Zeng, and Zhao (2020).

Panel A of Table4 reports the log marginal likelihoods, the posterior probabilities, and the ratios of the posterior probability to the prior probability of the top models. We further illustrate the posterior probabilities for all models in Figure1. The best combination of factors includes CRY, DUR, MOMs, and TERM. The model made up of these four factors yields the highest log marginal likelihood and, hence, the largest posterior probability. Thus, carry risk, duration risk, stock momentum, and term risk appear to be the most important factors in corporate bond markets in our sample. We call this set of factors the “no. 1 winning model,” or just “winning model.”

We view the model selection approach not only as a tool to determine the optimal model but also as one that helps us find the most important factors. Thus, it is also worth having a look at the next-best factor sets. In panel A of Table4, we report the top-four models based on a Bayes factor cutoff.13 The second-best factor model contains only 2 of the 4 factors of the winning model: CRY and MOMs. The third- and fourth-best models also include these two factors. They only differ in the additional factor included (DEF in case of the third-best and DUR in case of the fourth-best model).14

Thus, while quite naturally, based on a large set of candidate factors and a rather short sample period, the posterior probability of the winning model does not approach unity, a clear pattern emerges around the set of winning factors. This information is also reflected by the cumulative posterior probabilities of the factors presented in panel C of Table4. These are 100.0% for CRY, 99.96% for MOMs, 55.79% for DUR, and 47.78% for TERM. All other factors have cumulative posterior probabilities lower than 30%. Interestingly, the bond market factor yields the lowest cumulative posterior probability of 4.77%.

The DEF factor is only included in one of the top-four models and has a cumulative posterior probability of 29.20%. Thus, its explanatory power for the cross-section of corporate bond returns in our sample appears to be limited. This is surprising in light of the results of Gebhardt, Hvidkjaer, and Swaminathan (2005), who show that DEF betas perform well in explaining cross-sectional variation in beta-sorted portfolios. Thus, while performing well for these, the DEF factor appears to be much less able to explain the returns of other characteristics-sorted portfolios.15

Another metric to judge the performance of the selected models is the ratio of the posterior model probability to the prior model probability of any model $Mj$⁠, denoted by $Pr(Mj|y)Pr(Mj)$⁠. This ratio reflects the information improvement of the posterior over the prior, which is the same for all models, given the data observed. In the case of the winning model, improvement is very clear. Its posterior is more than 160 times as high as its prior.

In panel B of Table4, we also examine the performance of the existing factor models spanned by the set of factors that survive the first-step screening. We find that both the corporate bond CAPM and the FF3 model perform poorly. The posterior probabilities are 0.00% and the ratios of the posterior probability to the prior probability are 0.00. A model with all factors (which we consider as a benchmark by way of exception, despite the high correlations of MKTb, BVL, and DUR) also performs rather poorly with a posterior-to-prior-probability ratio of 0.02. This implies that all 11 candidate factors together contain information redundancies. Thus, in the trade-off between slightly enhanced in-sample performance and the parsimony encouraged by the Barillas and Shanken (2018) and Chib, Zeng, and Zhao (2020) model selection approach, adding all these additional factors hurts the model performance.

Thus, the most important set of factors in corporate bond markets appears to consist of CRY, DUR, MOMs, and TERM. Carry reflects the return of an asset if the market conditions stay the same (Koijen et al. 2018). As such, it is not unique to corporate bond markets. However, it is an important measure of risk and expected return. Duration and TERM are important due to the interest rate risk, which is a unique feature of bond markets that strongly differs from equity markets. Corporate bonds with higher interest rate risk earn systematically larger returns. Finally, high stock momentum increases the equity cushion available and reduces firm leverage, hence making the more senior claims of corporate bonds less risky. The factors associated with these corporate bond characteristics seem to systematically drive and explain corporate bond returns.

4. Asset Pricing Tests

4.1 Model Sharpe ratios

Having selected an optimal set of factors, we next turn to analyzing whether the selected winning models outperform other factor models based on more traditional model comparison approaches. That is, instead of Bayesian statistics, in this section, we use classical statistics and conduct pairwise tests of the equality of squared Sharpe ratios following Barillas et al. (2020). Their method enables us to provide reliable inference regarding relative model performance gauged by the squared Sharpe ratio improvement.

Table5 reports the differences between the sample squared Sharpe ratios (column model minus row model) of different pairs of models. The estimated model squared Sharpe ratios are modified to be unbiased in small samples. The associated p-values are shown in brackets.

The final column of Table5 clearly indicates that the top factor model of the model selection approach dominates all other existing models by producing a higher Sharpe ratio. The bias-adjusted squared Sharpe ratio of the winning model is higher by 0.45 compared to the CAPMbond model. For the other factor models, the improvements are generally only somewhat smaller. For example, compared to the FF3 model, the improvement is 0.43, compared to the augmented FF3 model 0.38, and compared to the BBW model 0.40. In terms of squared Sharpe ratios, the IPR and KPP models perform best among the existing models. However, the squared Sharpe ratio improvement of the no. 1 winning model is still 0.04 and 0.17, respectively. All these Sharpe ratio differences are highly statistically significant, as specified by the corresponding p-values that are virtually zero in all instances.

The no. 1 winning model also outperforms the second- to fourth-best models of the model selection. Its squared Sharpe ratios are significantly larger. The no. 2 to 4 winning models also outperform all other models except for the IPR model. Among the existing ones, the IPR model clearly performs best. This is not surprising as the model overlaps with the winning model in three of its factors.

Relying on comparisons of in-sample Sharpe ratios is not enough, though. Kan, Wang, and Zheng (2022) show that in the presence of estimation risk, the multifactor in-sample Sharpe ratios are typically unattainable for investors in real time. Therefore, we also analyze out-of-sample Sharpe ratios.

We present the results in Table6. As in Barillas and Shanken (2018), we show the full-sample Sharpe ratios of the models as well as the in- and out-of-sample Sharpe ratios for two different sample splitting schemes. Consistent with the results of Table5, the no. 1 winning model has the highest in-sample Sharpe ratio of 0.76, followed by the IPR model with 0.74 and the other winning models (0.67 up to 0.71).16